+
+
+def dual_face(K,F):
+ r"""
+ Return the dual face of ``F`` with respect to the cone ``K``.
+
+ OUTPUT:
+
+ A face of ``K.dual()``.
+
+ REFERENCES:
+
+ .. [HilgertHofmannLawson] Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie
+ D. Lawson. Lie groups, convex cones and semigroups. Oxford Mathematical
+ Monographs. Clarendon Press, Oxford, 1989. ISBN 9780198535690.
+
+ .. [Tam] Bit-Shun Tam. On the duality operator of a convex cone. Linear
+ Algebra and its Applications, 64:33-56, 1985, doi:10.1016/0024-3795(85)
+ 90265-4.
+
+ SETUP::
+
+ sage: from mjo.cone.faces import (dual_face, face_generated_by)
+
+ EXAMPLES:
+
+ The dual face of the first standard basis vector in three dimensions
+ is the face generated by the other two standard basis vectors::
+
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: F = Cone([(1,0,0)])
+ sage: dual_face(K,F).rays()
+ M(0, 0, 1),
+ M(0, 1, 0)
+ in 3-d lattice M
+
+ TESTS:
+
+ The dual face of ``K`` with respect to itself should be the
+ lineality space of its dual [Tam]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+ sage: K_dual = K.dual()
+ sage: lKd_gens = [ dir*l for dir in [1,-1] for l in K_dual.lines() ]
+ sage: linspace_K_dual = Cone(lKd_gens, K_dual.lattice())
+ sage: dual_face(K,K).is_equivalent(linspace_K_dual)
+ True
+
+ If ``K`` is proper, then the dual face of its trivial face is the
+ dual of ``K`` [Tam]_::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8,
+ ....: max_rays=10,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: L = K.lattice()
+ sage: trivial_face = Cone([L.zero()], L)
+ sage: dual_face(K,trivial_face).is_equivalent(K.dual())
+ True
+
+ The dual of the cone of ``K`` at ``y`` is the dual face of the face
+ of ``K`` generated by ``y`` ([Tam]_ Corollary 3.2)::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+ sage: y = K.random_element()
+ sage: S = [y]
+ sage: phi_y = face_generated_by(K,S)
+ sage: points_cone_gens = list(K.rays()) + [-z for z in phi_y.rays()]
+ sage: points_cone = Cone(points_cone_gens, K.lattice())
+ sage: points_cone.dual().is_equivalent(dual_face(K, phi_y))
+ True
+
+ Since all faces of a polyhedral cone are exposed, the dual face of a
+ dual face should be the original face [HilgertHofmannLawson]_::
+
+ sage: set_random_seed()
+ sage: def check_prop(K,F):
+ ....: return dual_face(K.dual(), dual_face(K,F)).is_equivalent(F)
+ sage: K = random_cone(max_ambient_dim=8, max_rays=10)
+ sage: all([check_prop(K,F) for F in K.face_lattice()])
+ True
+
+ """
+ # Ensure that F is actually a face of K before continuing.
+ if not F.is_face_of(K):
+ raise ValueError("%s is not a face of %s" % (F,K))
+
+ span_F = Cone([c*g for c in [1,-1] for g in F], F.lattice())
+ return K.dual().intersection(span_F.dual())